A kac moody verma module is not integrable in general

Fix a simple index $i$. By F5 its simple negative root space is one dimensional with specified nonzero vector $f_i$. Use the homogeneous basis in F4, replacing its vector at that degree by $f_i$ if needed; this is a single nonzero rescaling. For every $N\ge 0$, the ordered monomial consisting of $N$ copies of $f_i$ is a PBW basis monomial by F4, including the empty monomial $1$ at $N=0$. Hence these powers are nonzero and independent in $U({\mathfrak{n}}^{-})$. F3 identifies them with $f_i^N{v}_{\lambda }$, so none vanishes.

By 1.1, the single nonzero vector $v_{\lambda }$ from F1 is killed by no power of $f_i$. This fails the local nilpotence requirement F2 and proves nonintegrability. For the explicit witness $A=[2]$, $\lambda =0$, the label zero is dominant integral, while $v_0,f{v}_0,f^2{v}_0,\dots$ are all nonzero independent vectors. Thus the extra dominant-weight hypothesis does not repair the false claim. This includes the zeroth and first powers; there is no terminal power. A GCM with no simple indices is excluded, since then there is no lowering operator to witness failure. The basis construction and one specified rescaling require no AC.